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A generalized wave operator, i.e. a partially isometric operator defined by

$$ 
W _ {+} ( A _ {2} , A _ {1} )  =  s - \lim\limits _ {t \rightarrow x }  e ^ {it A _ {2} - it A _ {1} } P _ {1} ,
$$

where  $  A _ {1} $
and  $  A _ {2} $
are self-adjoint operators on a separable Hilbert space  $  H $, 
$  P _ {1} $
is an ortho-projector into  $  H _ {1,ac} $, 
and such that

$$ 
\{ {W _ {+} ( A _ {2} , A _ {1} ) x } : {
\| W _ {+} ( A _ {2} , A _ {1} ) x \| =
\| x \| } \}
 = \ 
H _ {2,ac} .
$$

Here  $  H _ {i,ac} $, 
$  i = 1, 2 $, 
is the set of all elements  $  x $
that are spectrally absolutely continuous with respect to  $  A _ {i} $, 
i.e. for which the spectral measure  $  \langle  E _ {A _ {i}  } ( \mu ) x, x \rangle $
of a set  $  M $
is absolutely continuous with respect to the Lebesgue measure  $  \mu $.

If the operator  $  W _ {+} ( A _ {2} , A _ {1} ) $, 
or the analogously defined operator  $  W _ {-} ( A _ {2} , A _ {1} ) $, 
exists and is complete, the  $  A _ {i,ac} $(
the parts of the operators  $  A _ {i} $
on  $  H _ {i,ac} $) 
are unitarily equivalent. If  $  A _ {1} $
and  $  A _ {2} $
are self-adjoint operators on  $  H $
and  $  A _ {2} = A _ {1} + c \langle  \cdot , f \rangle f $, 
where  $  f \in H $
and  $  c $
is real, then  $  W _  \pm  ( A _ {2} , A _ {1} ) $
and  $  W _  \pm  ( A _ {1} , A _ {2} ) $
exist and are complete.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Kato,   "Perturbation theory for linear operators" , Springer  (1966)  pp. Chapt. X Sect. 3</TD></TR></table>

====Comments====
An ortho-projector is usually called an orthogonal projector in the West.

An operator  $  W :  H \rightarrow H _ {1} $
is partially isometric if there is a closed linear subspace  $  M $
of  $  H $
such that  $  \| W u \| = \| u \| $
for  $  u \in M $
and  $  W v = 0 $
for  $  v \in M  ^  \perp  $, 
the orthogonal complement of  $  M $; 
the set  $  M $
is called the initial set of  $  W $
and  $  M _ {1} = W ( M) $
the final set of  $  W $.